Problem: What is the average value of $6x^2+8$ on the interval $[3,5]$ ?
Answer: In general, this is the average value of function $f$ over the interval $[a,b]$ : $\dfrac{\int_a^b f(x)\,dx}{b-a}$ In our case, ${f(x)=6x^2+8}$, ${a=3}$ and ${b=5}$ : $\begin{aligned} \dfrac{\int_{ a}^{ b} {f(x)}\,dx}{ b- a}&=\dfrac{\int_{{3}}^{ {5}} ({6x^2+8})\,dx}{{5}-{3}} \\\\ &=\dfrac{\Big[2x^3+8x\Big]_{3}^{5}}{2} \\\\ &=\dfrac{290-78}{2} \\\\ &=106 \end{aligned}$ In conclusion, the average value of $6x^2+8$ on the interval $[3,5]$ is $106$.